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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-hw1-review" class="internal"><span class="title">HW 1 Q5</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-19.html#notes-02-19-3" class="internal"><span class="title">Central Limit Theorem</span></a></div></li></ul>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-3" class="internal"><span class="title">More Central Limit Theorem</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-4" class="internal"><span class="title">Confidence Intervals</span></a></div></li>
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<section class="introduction" id="exam-02-2"><div class="para" id="exam-02-2-1">Show all relevant work.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="onepage" id="exam-02-3"><article class="exercise exercise-like" id="exam-02-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<div class="introduction" id="exam-02-3-1-1"><div class="para" id="exam-02-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="task exercise-like" id="exam-02-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="exam-02-3-1-2-1-1">Suppose <span class="process-math">\(X_1, \dotsc, X_n\)</span> are any random variables and <span class="process-math">\(S = (X_1 + X_2 + X_3 + \dotsb + X_n)^2\text{.}\)</span> Then <span class="process-math">\(S\)</span> is approximately normally distributed.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exam-02-3-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="exam-02-3-1-2-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 1.a.1"><a href="#exam-02-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.a"><a href="#exam-02-3-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="exam-02-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
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<div class="para" id="exam-02-3-1-3-1-1">Let <span class="process-math">\(Z\)</span> be the standard normal distribution. For any <span class="process-math">\(x\text{,}\)</span> <span class="process-math">\(\Pr(Z \geq x) = 1 - \Phi(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exam-02-3-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="exam-02-3-1-3-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.b.1"><a href="#exam-02-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.b"><a href="#exam-02-3-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="exam-02-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||
<div class="para" id="exam-02-3-1-4-1-1">If we’re using a set of data to find a confidence interval for a parameter, then the 95% confidence interval will be wider than the 90% confidence interval.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-02-3-1-4-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para" id="exam-02-3-1-4-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.c.1"><a href="#exam-02-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.c"><a href="#exam-02-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><article class="task exercise-like" id="exam-02-3-1-5"><h4 class="heading"><span class="codenumber">(d)</span></h4>
|
||
<div class="para" id="exam-02-3-1-5-1-1">In hypothesis testing, a type I error is the probability of accepting the null hypothesis.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-02-3-1-5-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
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<div class="para" id="exam-02-3-1-5-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.d.1"><a href="#exam-02-3-1-5-2" title="Copy heading and permalink for Solution 1.d.1" aria-label="Copy heading and permalink for Solution 1.d.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.d"><a href="#exam-02-3-1-5" title="Copy heading and permalink for Task 1.d" aria-label="Copy heading and permalink for Task 1.d">🔗</a></div></article><article class="task exercise-like" id="exam-02-3-1-6"><h4 class="heading"><span class="codenumber">(e)</span></h4>
|
||
<div class="para" id="exam-02-3-1-6-1-1">For a particular set of collected data, the <span class="process-math">\(p\)</span>-value of a 2-tailed test will be as large or larger than the <span class="process-math">\(p\)</span>-value of a 1-tailed test.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-02-3-1-6-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
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<div class="para" id="exam-02-3-1-6-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-6-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 1.e.1"><a href="#exam-02-3-1-6-2" title="Copy heading and permalink for Solution 1.e.1" aria-label="Copy heading and permalink for Solution 1.e.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.e"><a href="#exam-02-3-1-6" title="Copy heading and permalink for Task 1.e" aria-label="Copy heading and permalink for Task 1.e">🔗</a></div></article><article class="task exercise-like" id="exam-02-3-1-7"><h4 class="heading"><span class="codenumber">(f)</span></h4>
|
||
<div class="para" id="exam-02-3-1-7-1-1">In a <span class="process-math">\(\chi^2\)</span> test, if <span class="process-math">\(\chi^2 \lt 0.05\text{,}\)</span> then we can reject the null hypothesis.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exam-02-3-1-7-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para" id="exam-02-3-1-7-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-3-1-7-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Solution 1.f.1"><a href="#exam-02-3-1-7-2" title="Copy heading and permalink for Solution 1.f.1" aria-label="Copy heading and permalink for Solution 1.f.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 1.f"><a href="#exam-02-3-1-7" title="Copy heading and permalink for Task 1.f" aria-label="Copy heading and permalink for Task 1.f">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#exam-02-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><section class="onepage" id="exam-02-4"><article class="exercise exercise-like" id="exam-02-4-1"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="introduction" id="exam-02-4-1-1"><div class="para" id="exam-02-4-1-1-1">In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test. Indicate clearly which words or phrase in the description of the scenario would lead you to draw your conclusion.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div>
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<article class="task exercise-like" id="exam-02-4-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||
<div class="para" id="exam-02-4-1-2-1-1">We wonder whether the population of mice in a city have a different average weight to the population of mice in a rural town.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-02-4-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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||
<div class="para" id="exam-02-4-1-2-2-1">2-tailed, because we wonder whether the cities have "different average weight" without suspecting a direction of difference.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
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||
<div class="autopermalink" data-description="Solution 2.a.1"><a href="#exam-02-4-1-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 2.a"><a href="#exam-02-4-1-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="exam-02-4-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||
<div class="para" id="exam-02-4-1-3-1-1">A new fertilizer is designed to increase the yield of a particular species of fruit, and we wonder whether the fertilizer is effective.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-02-4-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para" id="exam-02-4-1-3-2-1">1-tailed, because the new fertilizer is "designed to increase the yield", so we suspect the change happens in a particular direction.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||
<div class="autopermalink" data-description="Solution 2.b.1"><a href="#exam-02-4-1-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
|
||
</div></details></div>
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||
<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 2.b"><a href="#exam-02-4-1-3" title="Copy heading and permalink for Task 2.b" aria-label="Copy heading and permalink for Task 2.b">🔗</a></div></article><article class="task exercise-like" id="exam-02-4-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||
<div class="para" id="exam-02-4-1-4-1-1">We find a weighted 6-sided die and suspect that it will roll a 6 more often than a fair die would.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exam-02-4-1-4-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para" id="exam-02-4-1-4-2-1">1-tailed, because we suspect the die will roll 6 "more often", indicating difference in a particular direction.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-4-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 2.c.1"><a href="#exam-02-4-1-4-2" title="Copy heading and permalink for Solution 2.c.1" aria-label="Copy heading and permalink for Solution 2.c.1">🔗</a></div>
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||
</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Task 2.c"><a href="#exam-02-4-1-4" title="Copy heading and permalink for Task 2.c" aria-label="Copy heading and permalink for Task 2.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#exam-02-4-1" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><section class="onepage" id="exam-02-5"><article class="exercise exercise-like" id="exam-02-5-1"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="para" id="exam-02-5-1-1-1">A coin has a probability <span class="process-math">\(\theta = 0.7\)</span> of coming up heads. We flip the coin 180 times, and write <span class="process-math">\(S\)</span> for the number of heads. Estimate <span class="process-math">\(\Pr(120 \leq S \leq 130)\text{.}\)</span> (Use a continuity correction if appropriate.)<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exam-02-5-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-5-1-2-1">
|
||
<div class="para">
|
||
<span class="process-math">\(S\)</span> is binomially distributed with <span class="process-math">\(\E(S) = (180)(0.7) = 126\)</span> and <span class="process-math">\(\Var(S) = (180)(0.7)(0.3) = 37.8\text{.}\)</span> Therefore:</div>
|
||
<div class="displaymath process-math" id="exam-02-5-1-2-1-4">
|
||
\begin{align*}
|
||
\Pr(120 \leq S \leq 130) \amp \approx \Pr\left( \frac{119.5 - 126}{\sqrt{37.8}} \leq Z \leq \frac{130.5 - 126}{\sqrt{37.8}}\right) \\
|
||
\amp \approx \Pr(-1.06 \leq Z \leq 0.73) \\
|
||
\amp = \Phi(0.73) - \Phi(-1.06) \\
|
||
\amp = \Phi(0.73) - \Phi(-1.06) \\
|
||
\amp \approx 0.7673 - 0.1446 \\
|
||
\amp = 0.6227.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 3.1"><a href="#exam-02-5-1-2" title="Copy heading and permalink for Solution 3.1" aria-label="Copy heading and permalink for Solution 3.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#exam-02-5-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="exam-02-5-2"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="para" id="exam-02-5-2-1-1">The weights of five apples are measured and recorded below. Find the sample mean, sample variance, and a 95% confidence interval around the sample mean for the weights of the apples. (Pretend that 5 measurements is enough for the CLT to apply. Do not use the <span class="process-math">\(t\)</span>-distribution.)<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines">Apple <span class="process-math">\(i\)</span>
|
||
</td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r1 l0 t0 lines">Weight <span class="process-math">\(W_i\)</span> (g)</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(140\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(120\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(165\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(168\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(137\)</span></td>
|
||
</tr>
|
||
</table></div>
|
||
<div class="solutions"><details id="exam-02-5-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-5-2-2-1">
|
||
<div class="displaymath process-math" id="exam-02-5-2-2-1-1">
|
||
\begin{align*}
|
||
A = \text{Avg weight} \amp = \frac{\sum W_i}{5} = 146 \\
|
||
\sum W_i^2 \amp = 108218 \\
|
||
s^2 \amp = \frac{\sum W_i^2 - n(A^2)}{n -1} = \frac{108218 - 5(146^2)}{4} = 409.5 \\
|
||
\sqrt{\frac{s^2}{n}} \amp = \sqrt{\frac{409.5}{5}} \approx 9.05 \\
|
||
\mu_{\ell} \amp = 146 - 1.96(9.05) = 128.262 \\
|
||
\mu_{h} \amp = 146 + 1.96(9.05) = 163.738
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 4.1"><a href="#exam-02-5-2-2" title="Copy heading and permalink for Solution 4.1" aria-label="Copy heading and permalink for Solution 4.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#exam-02-5-2" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="exam-02-6"><article class="exercise exercise-like" id="exam-02-6-1"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="para" id="exam-02-6-1-1-1">Suppose we’re told that a proportion of <span class="process-math">\(\theta = 0.4\)</span> of a population has a particular trait, but we suspect that this information is incorrect. We sample 160 people from the population and see 76 people with the trait. Is this enough evidence to reject the null hypothesis at the 0.05 significance level?<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-6-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exam-02-6-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-6-1-2-1">
|
||
<div class="para">"We suspect the information is incorrect" indicates a 2-tailed test. Let <span class="process-math">\(S\)</span> be the number of people in the sample with the trait. Then, under <span class="process-math">\(H_0\text{,}\)</span> <span class="process-math">\(S \sim \Bin(160, 0.4)\text{,}\)</span> so <span class="process-math">\(\E(S) = (160)(0.4) = 64\)</span> and <span class="process-math">\(\Var(S) = (160)(0.4)(0.6) = 38.4\text{.}\)</span> The observed <span class="process-math">\(76\)</span> people is <span class="process-math">\(12\)</span> more than expected. Equally extreme in the other direction would be <span class="process-math">\(12\)</span> fewer, so <span class="process-math">\(52\)</span> people. Therefore the <span class="process-math">\(p\)</span>-value is:</div>
|
||
<div class="displaymath process-math" id="exam-02-6-1-2-1-11">
|
||
\begin{align*}
|
||
\Pr(S \geq 76) + \Pr(S \leq 52) \amp \approx \Pr\left( Z \geq \frac{75.5 - 64}{\sqrt{38.4}} \right) + \Pr\left(Z \leq \frac{52.5 - 64}{\sqrt{38.4}}\right) \\
|
||
\amp \approx \Pr(Z \geq 1.86) + \Pr(Z \leq -1.86) \\
|
||
\amp = 2 \Phi(-1.86) \\
|
||
\amp = 2 (0.0314) \\
|
||
\amp = 0.0628 \gt 0.05,
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">so we do not reject <span class="process-math">\(H_0\text{.}\)</span>
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-6-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 5.1"><a href="#exam-02-6-1-2" title="Copy heading and permalink for Solution 5.1" aria-label="Copy heading and permalink for Solution 5.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#exam-02-6-1" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article></section><section class="onepage" id="exam-02-7"><article class="exercise exercise-like" id="exam-02-7-1"><h3 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="introduction" id="exam-02-7-1-1"><div class="para" id="exam-02-7-1-1-1">Suppose we have a coin that we’re told has probability <span class="process-math">\(\theta = 0.4\)</span> of coming up heads, but we suspect it comes up heads more often than that. We flip the coin 160 times.<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div></div>
|
||
<article class="task exercise-like" id="exam-02-7-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||
<div class="para" id="exam-02-7-1-2-1-1">What is the minimum number of heads we would need to see to reject the null hypothesis?<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exam-02-7-1-2-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-7-1-2-2-1">
|
||
<div class="para">"More often" suggests we should use a 1-tailed test. Let <span class="process-math">\(S\)</span> be the number of heads. Then, under <span class="process-math">\(H_0\text{,}\)</span> <span class="process-math">\(S \sim \Bin(160, 0.4)\text{,}\)</span> so <span class="process-math">\(\E(S) = (160)(0.4) = 64\)</span> and <span class="process-math">\(\Var(S) = (160)(0.4)(0.6) = 38.4\text{.}\)</span> Therefore, the <span class="process-math">\(p\)</span>-value when observing <span class="process-math">\(k\)</span> heads would be:</div>
|
||
<div class="displaymath process-math" id="exam-02-7-1-2-2-1-8">
|
||
\begin{align*}
|
||
\Pr(S \geq k) \amp \approx \Pr\left( Z \geq \underbrace{\frac{k - 0.5 - 64}{\sqrt{38.4}}}_{z} \right) \\
|
||
\Pr(Z \geq z) \amp = 0.05 \\
|
||
1 - \Phi(z) \amp = 0.05 \quad \Rightarrow \quad \Phi(z) = 0.95.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">We would need the <span class="process-math">\(z\)</span>-score to be at least 1.65 to reject <span class="process-math">\(H_0\text{,}\)</span> so:</div>
|
||
<div class="displaymath process-math" id="exam-02-7-1-2-2-1-11">
|
||
\begin{align*}
|
||
\frac{k - 0.5 - 64}{\sqrt{38.4}} \amp = 1.65 \\
|
||
k = 64.5 + 1.65 \sqrt{38.4} \amp \approx 74.72
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Therefore, it would take at least <span class="process-math">\(75\)</span> heads to reject <span class="process-math">\(H_0\text{.}\)</span>
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 6.a.1"><a href="#exam-02-7-1-2-2" title="Copy heading and permalink for Solution 6.a.1" aria-label="Copy heading and permalink for Solution 6.a.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Task 6.a"><a href="#exam-02-7-1-2" title="Copy heading and permalink for Task 6.a" aria-label="Copy heading and permalink for Task 6.a">🔗</a></div></article><article class="task exercise-like" id="exam-02-7-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||
<div class="para" id="exam-02-7-1-3-1-1">If the coin is actually fair, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exam-02-7-1-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-7-1-3-2-1">
|
||
<div class="para">If the coin is fair, then <span class="process-math">\(\E(S) = (160)(0.5) = 80\)</span> and <span class="process-math">\(\Var(S) = (160)(0.5)(0.5) = 40\text{.}\)</span> The probability of rejecting <span class="process-math">\(H_0\)</span> is then:</div>
|
||
<div class="displaymath process-math" id="exam-02-7-1-3-2-1-4">
|
||
\begin{align*}
|
||
\Pr(S \geq 75) \amp \approx \Pr\left(Z \geq \frac{74.5 - 80}{\sqrt{40}}\right) \\
|
||
\amp \approx \Pr(Z \geq -0.87) \\
|
||
\amp \approx 1 - \Phi(-0.87) \\
|
||
\amp \approx 1 - 0.1922 \\
|
||
\amp = 0.8078.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 6.b.1"><a href="#exam-02-7-1-3-2" title="Copy heading and permalink for Solution 6.b.1" aria-label="Copy heading and permalink for Solution 6.b.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Task 6.b"><a href="#exam-02-7-1-3" title="Copy heading and permalink for Task 6.b" aria-label="Copy heading and permalink for Task 6.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 6"><a href="#exam-02-7-1" title="Copy heading and permalink for Worksheet Exercise 6" aria-label="Copy heading and permalink for Worksheet Exercise 6">🔗</a></div></article></section><section class="onepage" id="exam-02-8"><article class="exercise exercise-like" id="exam-02-8-1"><h3 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||
<div class="para" id="exam-02-8-1-1-1">A store owner assumes an equal number of people on average come into the store each day of the week. Are the following counts of 400 observed shoppers consistent with this hypothesis?<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-8-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines">Day</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Mon</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Tue</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Wed</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Thu</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Fri</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r1 l0 t0 lines">Observed Shoppers</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(83\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(61\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(65\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(90\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(101\)</span></td>
|
||
</tr>
|
||
</table></div>
|
||
<div class="solutions"><details id="exam-02-8-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="exam-02-8-1-2-1">
|
||
<div class="para">Under the null hypothesis, we would expect <span class="process-math">\(400/5 = 80\)</span> people each day. So:</div>
|
||
<div class="displaymath process-math" id="exam-02-8-1-2-1-2">
|
||
\begin{align*}
|
||
\chi^2 \amp = \frac{(83 - 80)^2}{80} + \dotsb + \frac{(101 - 80)^2}{80} = 14.2.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">The number of degrees of freedom is <span class="process-math">\((5 - 1) - (0) = 4\text{,}\)</span> so the critical value is <span class="process-math">\(9.488\text{.}\)</span> Since <span class="process-math">\(14.2 \gt 9.488\text{,}\)</span> we reject the null hypothesis.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-8-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 7.1"><a href="#exam-02-8-1-2" title="Copy heading and permalink for Solution 7.1" aria-label="Copy heading and permalink for Solution 7.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="workspace" data-space="1in"></div>
|
||
<div class="autopermalink" data-description="Worksheet Exercise 7"><a href="#exam-02-8-1" title="Copy heading and permalink for Worksheet Exercise 7" aria-label="Copy heading and permalink for Worksheet Exercise 7">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Exam 2"><a href="#exam-02" title="Copy heading and permalink for Worksheet: Exam 2" aria-label="Copy heading and permalink for Worksheet: Exam 2">🔗</a></div></section>
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